Determination of the Genus of Surfaces from the Spectrum of Schrödinger Operators attached to height functions. (An inverse spectral problem for surfaces)

Brice Camus

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arXiv · 1303.1866

Determination of the Genus of Surfaces from the Spectrum of Schrödinger Operators attached to height functions. (An inverse spectral problem for surfaces)

Brice Camus

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Abstract

Using results on inverse spectral problems, in particular the so-called new wave invariants attached to a classical equilibrium, we show that it is possible to determine the Morse index of height functions. For compact Riemannian surfaces $M\subset \mathbb{R}^3$ this imply that we can retrieve the topology (via the genus). Our results are independent from the choice of a metric on $M$ and can be obtained from the choice of a 'generic' height-function. For surfaces of genus zero, diffeomorphic to a 2-sphere, the method allows to detect the convexity, or the local convexity of the surface. Keywords : Micro-local analysis; Schrödinger operators; Inverse spectral problems.

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Brice Camus. 2013-03-08. Determination of the Genus of Surfaces from the Spectrum of Schrödinger Operators attached to height functions. (An inverse spectral problem for surfaces). https://arxiv.org/abs/1303.1866

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